# 2.8 The angles of a triangle

The sum of the angles of any triangle is 180°. This property can be demonstrated in several ways. One way is to draw a triangle on a piece of paper, mark each angle with a different symbol, and then cut out the angles and arrange them side by side, touching one another as illustrated.

You can see *why* it is that the angles fit together in this way by looking at the triangle below. An extra line has been added parallel to the base. The angle of the triangle, , is equal to the angle *β* at the top (they are alternate angles), and similarly the angle of the triangle, , is equal to the angle *γ* at the top (they are also alternate angles). The three angles at the top (*β*, *γ* and the angle of the triangle, ) form a straight line of total angle 180°, and so the angles of the triangle must also add up to 180°.

The sum of the angles of a triangle is 180°.

The fact that the angles of a triangle add up to 180° is another angle property that enables you to find unknown angles.

## Example 7

Find *α*, *β* and *θ* in the diagram below.

### Answer

First, look at the angles of Δ*ABD*: and

Then, by the angle sum property of triangles,

So

and

As *CDB* is a straight line and *α* = 60°, it follows that

Now consider the angles of Δ*ADC*: and

Therefore

So

(Check for yourself that the angles of Δ*ABC* also add up to 180°.)

It is possible to deduce more information about the angles in certain special kinds of triangles.

In a *right-angled triangle*, since one angle is a right angle (90°), the other two angles must add up to 90°. Thus, in the example below, *α* + *β* = 90°.

In an *equilateral triangle*, all the angles are the same size. So each angle of an equilateral triangle must be 180° ÷ 3 = 60°.

In an *isosceles triangle*, two sides are of equal length and the angles opposite those sides are equal. Therefore, *α* = *β* in the triangle below.

Such angles are often called **base angles**.

This means that there are only two different sizes of angle in an isosceles triangle: if the size of one angle is known, the sizes of the other two angles can easily be found. The next example shows how this is done.

## Example 8

Find the unknown angles in these isosceles triangles, which represent parts of the roof supports of a house.

### Answer

(a) As

*α*and 50° are the base angles,*α*= 50°. By the angle sum property of triangles,therefore

(b) As

*γ*and*δ*are the base angles,*γ*=*δ*. In this triangle,therefore

The various angle properties can also be used to find the sum of the angles of a quadrilateral.

## Example 9

The diagram below represents the four stages of a walk drawn on an Ordnance Survey map.

The figure *ABCD* is a quadrilateral. Find *θ* and *φ*, and thus the sum of all the angles of the quadrilateral.

### Answer

From Δ*ABC*,

From Δ*ACD*,

Then the sum of all the angles of the quadrilateral is

In fact, you can find the sum of the four angles of a quadrilateral without calculating each angle as in Example 9. Look again at the quadrilateral: the dotted line splits it into two triangles, and the angles of these triangles together make up the angles of the quadrilateral. Each triangle has an angle sum of 180°, so the angle sum of the quadrilateral is 2 × 180° = 360°. This is true for *any* quadrilateral.

The sum of the angles of a quadrilateral is 360°.

Similarly, other polygons (that is, other shapes with straight sides) can be divided into triangles to find the sum of their angles.